Correction of NMR artifacts due to constant-velocity axial motion and spin-lattice relaxation

ABSTRACT

NMR spin echo signals are corrected for axial motion of the borehole logging tool. An additional correction may be applied to correct for incomplete polarization of nuclear spins due to an insufficient wait time between pulse sequences.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention relates generally to determining geologicalproperties of subsurface formations using Nuclear Magnetic Resonance(“NMR”) methods for logging wellbores, particularly for correcting forthe effects of tool motion and pulse sequence timing on NMR signals.

2. Description of the Related Art

A variety of techniques are currently utilized in determining thepresence and estimation of quantities of hydrocarbons (oil and gas) inearth formations. These methods are designed to determine formationparameters, including among other things, the resistivity, porosity andpermeability of the rock formation surrounding the wellbore drilled forrecovering the hydrocarbons. Typically, the tools designed to providethe desired information are used to log the wellbore. Much of thelogging is done after the well bores have been drilled. More recently,wellbores have been logged while drilling, which is referred to asmeasurement-while-drilling (MWD) or logging-while-drilling (LWD).

One commonly used technique involves utilizing Nuclear MagneticResonance (NMR) logging tools and methods for determining, among otherthings, porosity, hydrocarbon saturation and permeability of the rockformations. The NMR logging tools are utilized to excite the nuclei ofthe liquids in the geological formations surrounding the wellbore sothat certain parameters such as nuclear spin density, longitudinalrelaxation time (generally referred to in the art as T₁) and transverserelaxation time (generally referred to as T₂) of the geologicalformations can be measured. From such measurements, porosity,permeability and hydrocarbon saturation are determined, which providesvaluable information about the make-up of the geological formations andthe amount of extractable hydrocarbons.

The NMR tools generate a static magnetic field in a region of interestsurrounding the wellbore. NMR is based on the fact that the nuclei ofmany elements have angular momentum (spin) and a magnetic moment. Thenuclei have a characteristic Larmor resonant frequency related to themagnitude of the magnetic field in their locality. Over time the nuclearspins align themselves along an externally applied static magnetic fieldcreating a net magnetization. This equilibrium situation can bedisturbed by a pulse of an oscillating magnetic field, which tips thespins with resonant frequency within the bandwidth of the oscillatingmagnetic field away from the static field direction. The angle θ throughwhich the spins exactly on resonance are tipped is given by theequation:θ=γB ₁ t _(p)/2  (1)where γ is the gyromagnetic ratio, B₁ is the magnetic flux densityamplitude of the sinusoidally oscillating field and t_(p) is theduration of the RF pulse.

After tipping, the spins precess around the static field at a particularfrequency known as the Larmor frequency ω₀ given byω₀=γB₀  (2)where B₀ is the static magnetic flux density. For hydrogen nuclei γ/2π=4258 Hz/Gauss, so that a static field of 235 Gauss, would produce aprecession frequency of 1 MHz. At the same time, the magnetizationreturns to the equilibrium direction (i.e., aligned with the staticfield) according to a decay time known as the “spin-lattice relaxationtime” or T₁. T₁ is controlled by the molecular environment and istypically ten to one thousand milliseconds in rocks.

At the end of a θ=90° tipping pulse, spins on resonance are pointed in acommon direction perpendicular to the static field, and they precess atthe Larmor frequency. However, because of inhomogeneity in the staticfield due to the constraints on tool shape, imperfect instrumentation,or microscopic material heterogeneities, each nuclear spin precesses ata slightly different rate. Hence, after a time long compared to theprecession period, but shorter than T₁, the spins will no longer beprecessing in phase. This de-phasing occurs with a time constant that iscommonly referred to as T₂ *. Dephasing due to static fieldinhomogeneity can be recovered by generating spin echoes (see below).The remaining dephasing is characterized by the time constant T₂ and isdue to properties of the material.

A receiving coil is designed so that a voltage is induced by theprecessing spins. Only that component of the nuclear magnetization thatis precessing in the plane perpendicular to the static field is sensedby the coil. After a 180° tipping pulse (an “inversion pulse”), thespins on resonance are aligned opposite to the static field and themagnetization relaxes along the static field axis to the equilibriumdirection. Hence, a signal will be generated after a 90° tipping pulse,but not after a 180° tipping pulse in a generally uniform magneticfield.

While many different methods for measuring T₁ have been developed, asingle standard known as the CPMG sequence (Carr-Purcell-Meiboom-Gill)for measuring T₂ has evolved. In contrast to laboratory NMR magnets,well logging tools have inhomogeneous magnetic fields due to theconstraints on placing the magnets within a tubular tool and theinherent “inside-out” geometry. Maxwell's divergence theorem dictatesthat there cannot be a region of high homogeneity outside the tool.Therefore in typical well bores, T₂*<<T₂, and the free induction decaybecomes a measurement of the apparatus-induced inhomogeneities. Tomeasure the true T₂ in such situations, it is necessary to cancel theeffect of the apparatus-induced inhomogeneities. To accomplish the same,a series of pulses is applied to repeatedly refocus the spin system,canceling the T₂* effects and forming a series of spin echoes. The decayof echo amplitude is a true measure of the decay due to materialproperties. Furthermore it can be shown that the decay is in factcomposed of a number of different decay components forming a T₂distribution. The echo decay data can be processed to reveal thisdistribution which is related to rock pore size distribution and otherparameters of interest to the well log analyst.

Tool motion can seriously affect the performance of NMR tools used in anMWD environment. NMR tools that have static magnetic fields withcomplete rotational symmetry are unaffected by rotation of the toolsince the fields in the region of examination do not change during themeasurement sequence. However, any axial or transverse (orthogonal tothe tool axis) component of tool motion due to vibration will affect theNMR signal.

There are many well-known artifacts of motion that show up in signals indownhole logging. These artifacts are theoretically expected and areattributable to such factors as rotation, transverse vibration and axialmotion. In addition to these motion artifacts the NMR signal amplitudecan be reduced due to insufficient wait time for polarization after theend of an echo sequence.

Artifacts of rotation are a result of the typical stationary B₀ field ofthe system not being completely axisymmetric. Rotation of thedrillstring therefore causes (periodic) NMR signal losses during a spinecho train. Artifacts from transverse vibrations generally occur becauseof drilling or because of mud circulation through a mud motor. Theobtained vibration frequency spectrum usually includes some dominantfrequencies that are directly related to the rotational speed of themotor or drill string.

Axial motion of the drill string gives rise to two distinct artifacts. Afirst artifact of axial motion is caused by the motion of thedrillstring through the borehole. The rate of penetration (ROP) of thedrill string can be recorded electronically and later retrieved from acomputer file. Obviously, the ROP at the drill bit differs slightly fromthe electronically-recorded value of the ROP which is measured at thesurface. This discrepancy between ROP values is due to limited timeresolution of the computer file as well as to flexibility of the drillstring. A second artifact of axial motion exhibits itself as higherfrequency axial vibrations. These vibrations can be measured with anaccelerometer in the NMR tool.

Application of an insufficient wait time between consecutive pulsesequences can give rise to yet another artifact. Typically, after theend of an echo sequence obtained with axial motion of the drill string,the z-magnetization is substantially zero. This z-magnetization isgenerally non-zero when no such motion exists. A wait time is generallyapplied after an echo sequence to allow the protons to re-align alongthe direction of the static magnetic field. This re-magnetization occurswith a characteristic relaxation time known as the spin-latticerelaxation time T₁. Usually, there exists a distribution of T₁ timessimilar to the well-known T₂ distribution.

U.S. Pat. No. 5,389,877 issued to Sezginer describes a truncated CPMGsequence in which the sequence duration and recovery delay are so shortthat only signals from the clay and capillary bound fluids are detected.A truncated sequence has the advantage that the effect of tool motion onthe measurements is reduced due to the short measurement time (approx.50 ms, compared to greater than 300 ms for normal downhole CPMGmeasurements.) As discussed in U.S. Pat. No. 5,705,927 issued toKleinberg, resonance regions of many prior art instruments are of theorder of 1 mm. Accordingly, a lateral vibration at a frequency of 50 Hzhaving an amplitude of 1 mm (10 g acceleration) would disable theinstrument. The Kleinberg '927 patent discloses making the length ofeach CPMG sequence small, e.g. 10 ms, so that the drill collar cannot bedisplaced by a significant fraction of the vertical or radial extent ofthe sensitive region during a CPMG pulse sequence. However, as notedabove, using such short sequences and short wait times only gives anindication of the bound fluid volume and gives no indication of thetotal fluid volume.

U.S. Pat. No. 6,268,726 to Prammer et al., teaches the use of motionsensors on an MWD apparatus that makes measurements of tool motion of aNMR sensor assembly. Measurements are made by the NMR sensor duringcontinued drilling operations, and subsequently, the measurements madeby the motion sensor are used to select a subset of the NMR measurementsthat meet certain requirements on tool motion and hence would beexpected to give a reasonable insensitivity to tool motion. U.S. Pat.No. 6,459,263 to Hawkes et al, having the same assignee as the presentapplication and the contents of which are fully incorporated herein byreference, uses the output of motion sensors in combination withpredictive filtering to control the timing of pulses for a modified (asin the Hawkes '013 patent) or conventional CPMG sequence.

U.S. Pat. No. 6,566,874 to Speier et al. teaches several approaches todealing with problems associated with tool motion. In one embodiment,measurements are made of two different echo trains that have differentsensitivities to tool motion. A tool is used having two differentregions of examination: a high gradient zone defined by one set ofmagnets and antennas, and a low gradient zone defined by another set ofmagnets and antennas. The effect of tool motion on the signal amplitudeis greater in the high gradient zone than in the low gradient zone.Using these two sets of signals and knowing the gradients of therespective zones, it is possible to estimate what the signal would havebeen without the tool motion. The Speier '874 patent also teaches thatsensitivity to motion may be varied by different field geometries withdifferent gradients. This requirement of having two different regions ofexamination complicates the hardware. Another drawback (noted in Speier'874) to the above-described techniques is that the measurements must beseparated in time and/or space. In order to interpret the results it isassumed that, in the absence of motion, the NMR signal (and thereforethe formation measured) is the same in both measurements. For acontinuously moving logging tool, this condition is not always given.Also the motion during the two measurements should be the same, or atleast have the same characteristics.

In another embodiment taught by Speier '874, measurements are processedto obtain both the T₁ and T₂ distribution. The effect of tool motion isdifferent on the two types of measurements. This approach has at leasttwo drawbacks. The first is that T₁ determination is time consuming. Asecond drawback is that in the absence of an exact knowledge of theratio of T₁/T₂ , the method can only be used for quality control and notfor determining both the T₁ and T₂ distributions.

There is a need for computational methods to reduce the effects ofmotion artifacts encountered in MWD testing. The method should correctNMR signals due to (constant) ROP as well as insufficient wait time inMWD testing. This method should be combinable with other methods ofmotion corrections or methods that reduce motion artifacts. The methodshould be usable with standard pulse sequences in the prior art.

SUMMARY OF THE INVENTION

The present invention is a method of processing and an apparatus usedfor processing Nuclear Magnetic Resonance (NMR) signals from an earthformation. The NMR tool is conveyed into a borehole in the earthformation and moved with an axial velocity in the borehole. Nuclearspins in the earth formations are polarized. An antenna on the NMRlogging tool is pulsed with a pulse sequence to produce spin echosignals. The pulse sequence includes an excitation pulse and a pluralityof refocusing pulses. The spin echo signals are corrected using afunction of the velocity to give corrected spin echo signals. The NMRlogging tool may be conveyed into the borehole on a wireline, slickline,drillstring, or coiled tubing. The correction may be implemented byscaling the spin echo signals by a normalizing function related to theaxial velocity and a reference velocity. The correction factor may beapplied to the in-phase component, quadrature component or to theamplitude of the spin echo signals. The first correction compensates forone type of effects caused by tool motion.

In one embodiment of the invention, a plurality of pulse sequences isapplied with a wait time therebetween. A second correction may beapplied to compensate for the excessive premagnetization and also forinsufficient wait time that would not allow full magnetization of thenuclei used for NMR. This second correction may be in addition to orapplied independently of the first correction. The second correction iscalculated for the longitudinal relaxation values corresponding to thebins of the T₂ distribution. Correction B is then executed bymultiplying each bin of the T₂ distribution by its correction B factor.The result is a T2 distribution where constant axial velocity artifactsand insufficient recovery time artifacts have been eliminated.

BRIEF DESCRIPTION OF THE DRAWINGS

The present invention is best understood with reference to theaccompanying figures in which like numerals refer to like elements andin which:

FIG. 1 (Prior Art) shows a measurement-while-drilling tool suitable foruse with the present invention;

FIG. 2 (Prior Art) shows a sensor section of ameasurement-while-drilling device suitable for use with the presentinvention;

FIG. 3 shows a typical pulse sequence usable with the present invention;

FIG. 4 shows six NMR spin echo decay curves obtained at varying axialmotions of a drillstring through a formation;

FIG. 5 shows the results of a compression of the horizontal axis of FIG.4;

FIG. 6 shows simulations obtained with an infinite T₂ decay;

FIG. 7 shows simulations using an RF current amplitude different fromthat used in FIG. 6;

FIG. 8 shows the effect of applying a correction function of the presentinvention for a simulation with finite T₂;

FIG. 9 shows 2 ORPS sequences separated by a wait time TW;

FIG. 10 shows a laboratory example of echo decays produced using thesecond ORPS sequence from FIG. 9;

FIG. 11 shows the use of an aperiodic pulse sequence (APS);

FIG. 12 shows the effect of hyperpolarization and insufficient waittime;

FIG. 13 shows the error of fitted correction B;

FIG. 14 shows a flowchart of the present invention in the presence of aT₁ distribution; and

FIG. 15 shows a simulation of NMR data with the use of an APS sequence(FIG. 11) and applied corrections A and B.

DETAILED DESCRIPTION OF THE INVENTION

FIG. 1 shows a schematic diagram of a drilling system 10 with adrillstring 20 carrying a drilling assembly 90 (also referred to as thebottom hole assembly, or “BHA”) conveyed in a “wellbore” or “borehole”26 for drilling the wellbore. The drilling system 10 includes aconventional derrick 11 erected on a floor 12 which supports a rotarytable 14 that is rotated by a prime mover such as an electric motor (notshown) at a desired rotational speed. The drillstring 20 includes atubing such as a drill pipe 22 or a coiled-tubing extending downwardfrom the surface into the borehole 26. The drillstring 20 is pushed intothe wellbore 26 when a drill pipe 22 is used as the tubing. Forcoiled-tubing applications, a tubing injector, such as an injector (notshown), however, is used to move the tubing from a source thereof, suchas a reel (not shown), to the wellbore 26. The drill bit 50 attached tothe end of the drillstring breaks up the geological formations when itis rotated to drill the borehole 26. If a drill pipe 22 is used, thedrillstring 20 is coupled to a drawworks 30 via a Kelly joint 21, swivel28, and line 29 through a pulley 23. During drilling operations, thedrawworks 30 is operated to control the weight on bit, which is animportant parameter that affects the rate of penetration. The operationof the drawworks is well known in the art and is thus not described indetail herein. For the purposes of this invention, it is necessary toknow the axial velocity (rate of penetration or ROP) of the bottomholeassembly. Depth information and ROP may be communicated downhole from asurface location. Alternatively, the method disclosed in U.S. Pat. No.6,769,497 to Dubinsky et al. having the same assignee as the presentapplication and the contents of which are incorporated herein byreference may be used. The method of Dubinsky uses axial accelerometersto determine the ROP. During drilling operations, a suitable drillingfluid 31 from a mud pit (source) 32 is circulated under pressure througha channel in the drillstring 20 by a mud pump 34. The drilling fluidpasses from the mud pump 34 into the drillstring 20 via a desurger (notshown), fluid line 38 and Kelly joint 21. The drilling fluid 31 isdischarged at the borehole bottom 51 through an opening in the drill bit50. The drilling fluid 31 circulates uphole through the annular space 27between the drillstring 20 and the borehole 26 and returns to the mudpit 32 via a return line 35. The drilling fluid acts to lubricate thedrill bit 50 and to carry borehole cutting or chips away from the drillbit 50. A sensor S₁ typically placed in the line 38 provides informationabout the fluid flow rate. A surface torque sensor S₂ and a sensor S₃associated with the drillstring 20 respectively provide informationabout the torque and rotational speed of the drillstring. Additionally,a sensor (not shown) associated with line 29 is used to provide the hookload of the drillstring 20.

In one embodiment of the invention, the drill bit 50 is rotated by onlyrotating the drill pipe 22. In another embodiment of the invention, adownhole motor 55 (mud motor) is disposed in the drilling assembly 90 torotate the drill bit 50 and the drill pipe 22 is rotated usually tosupplement the rotational power, if required, and to effect changes inthe drilling direction.

In an exemplary embodiment of FIG. 1, the mud motor 55 is coupled to thedrill bit 50 via a drive shaft (not shown) disposed in a bearingassembly 57. The mud motor rotates the drill bit 50 when the drillingfluid 31 passes through the mud motor 55 under pressure. The bearingassembly 57 supports the radial and axial forces of the drill bit. Astabilizer 58 coupled to the bearing assembly 57 acts as a centralizerfor the lowermost portion of the mud motor assembly.

In one embodiment of the invention, a drilling sensor module 59 isplaced near the drill bit 50. The drilling sensor module containssensors, circuitry and processing software and algorithms relating tothe dynamic drilling parameters. Such parameters typically include bitbounce, stick-slip of the drilling assembly, backward rotation, torque,shocks, borehole and annulus pressure, acceleration measurements andother measurements of the drill bit condition. A suitable telemetry orcommunication sub 72 using, for example, two-way telemetry, is alsoprovided as illustrated in the drilling assembly 90. The drilling sensormodule processes the sensor information and transmits it to the surfacecontrol unit 40 via the telemetry system 72.

The communication sub 72, a power unit 78 and an MWD tool 79 are allconnected in tandem with the drillstring 20. Flex subs, for example, areused in connecting the MWD tool 79 in the drilling assembly 90. Suchsubs and tools form the bottom hole drilling assembly 90 between thedrillstring 20 and the drill bit 50. The drilling assembly 90 makesvarious measurements including the pulsed nuclear magnetic resonancemeasurements while the borehole 26 is being drilled. The communicationsub 72 obtains the signals and measurements and transfers the signals,using two-way telemetry, for example, to be processed on the surface.Alternatively, the signals can be processed using a downhole processorin the drilling assembly 90.

The surface control unit or processor 40 also receives signals fromother downhole sensors and devices and signals from sensors S₁-S₃ andother sensors used in the system 10 and processes such signals accordingto programmed instructions provided to the surface control unit 40. Thesurface control unit 40 displays desired drilling parameters and otherinformation on a display/monitor 42 utilized by an operator to controlthe drilling operations. The surface control unit 40 typically includesa computer or a microprocessor-based processing system, memory forstoring programs or models and data, a recorder for recording data, andother peripherals. The control unit 40 is typically adapted to activatealarms 44 when certain unsafe or undesirable operating conditions occur.

A suitable device for use of the present invention is disclosed in U.S.Pat. No. 6,215,304 to Slade, the contents of which are fullyincorporated herein by reference. It should be noted that the devicetaught by Slade is for exemplary purposes only, and the method of thepresent invention may be used with many other NMR logging devices, andmay be used for wireline as well as MWD applications. Examples of suchdevices are given in U.S. Pat. No. 5,557,201 to Kleinberg, U.S. Pat. No.5,280,243 to Miller, U.S. Pat. No. 5,055,787 to Kleinberg, and U.S. Pat.No. 5,698,979 to Taicher.

Referring now to FIG. 2, the tool has a drill bit 107 at one end, asensor section 102 behind the drill head, and electronics 101. Thesensor section 102 comprises a magnetic field generating assembly forgenerating a B₀ magnetic field (which is substantially time invariantover the duration of a measurement), and an RF system for transmittingand receiving RF magnetic pulses and echoes. The magnetic fieldgenerating assembly comprises a pair of axially spaced main magnets 103,104 having opposed pole orientations (i.e. with like magnetic polesfacing each other), and three ferrite members 109, 110 axially arrangedbetween the magnets 103, 104. The ferrite members are made of “soft”ferrite which can be distinguished over “hard” ferrite by the shape ofthe BH curve which affects both intrinsic coercivity (H_(j) theintersection with the H axis) and initial permeability (μ_(i), thegradient of the BH curve in the unmagnetized case). Soft ferrite μ_(i)values typically range from 10 to 10000 whereas hard ferrite has μ_(i),of about 1. Therefore the soft ferrite has large initial permeability(typically greater than 10, preferably greater than 1000). The RF systemcomprises a set of RF transmit antenna and RF receive antenna coilwindings 105 arranged as a central “field forming” solenoid group 113and a pair of outer “coupling control” solenoid groups 114.

The tool has a mud pipe 160 with a clear central bore 106 and a numberof exit apertures 161-164 to carry drilling mud to the bit 107, and themain body of the tool is provided by a drill collar 108. Drilling mud ispumped down the mud pipe 160 by a pump 121 returning around the tool andthe entire tool is rotated by a drive 120. Coiled tubing or adrillstring may be used for coupling the drive to the downhole assembly.

The drill collar 108 provides a recess 170 for RF transmit antenna andRF receive antenna coil windings 105. Gaps in the pockets between thesoft ferrite members are filled with non-conducting material 131, 135(e.g: ceramic or high temperature plastic) and the RF coils 113, 114 arethen wound over the soft ferrite members 109, 110. The soft ferrites109, 110 and RF coil assembly 113, 114 are pressure impregnated withsuitable high temperature, low viscosity epoxy resin (not shown) toharden the system against the effects of vibration, seal againstdrilling fluid at well pressure, and reduce the possibility ofmagnetoacoustic oscillations. The RF coils 113, 114 are then coveredwith wear plates 111 typically ceramic or other durable non-conductingmaterial to protect them from the rock chippings flowing upwards pastthe tool in the borehole mud.

Because of the opposed magnet configuration, the device of Slade has anaxisymmetric magnetic field and region of investigation 112 that isunaffected by tool rotation. Use of the ferrite results in a region ofinvestigation that is close to the borehole. This is not a major problemon a MWD tool because there is little invasion of the formation byborehole drilling fluids prior to the logging. The region ofinvestigation is within a shell with a radial thickness of about 20 mmand an axial length of about 50 mm. The gradient within the region ofinvestigation is less than 2.7 G/cm. It is to be noted that these valuesare for the Slade device and, as noted above, the method of the presentinvention may also be used with other suitable NMR devices.

Two magnetic fields are used to conduct a typical NMR measurement: astatic magnetic field B₀ and an alternating magnetic field B₁ having acomponent orthogonal to B₀. Pulsed NMR is used in which the alternatingfield B₁ is radiated into the sample as a sequence of bursts (usuallycalled pulses). A typical pulse sequence is shown in FIG. 3. The B₁pulse sequence comprises an excitation pulse 200 followed by a pluralityof refocusing pulses (202 a, 202 b, 202 c, 202 d, 202 e, . . . ). Spinechoes depicted by 205 a, 205 b, 205 c, 205 d, 205 e, . . . form betweenthese refocusing pulses. These echoes manifest themselves as rotatingmacroscopic magnetizations and can be detected with a receiver coil. Theinduced voltages/currents in this coil are the desired NMR signals. Inorder to obtain NMR signals and refocus them correctly, it is importantto adhere to NMR resonance conditions, i.e. B₀ and B₁ amplitudes as wellas pulse phases and shapes need to be chosen correctly as known topeople familiar with the art of NMR (see Fukushima, Experimental PulseNMR: A Nuts and Bolts Approach, 1981, Tenth printing, January 1998.). Anexemplary optimized echo sequence called ORPS is discussed, for example,in Hawkes '013. In the ORPS sequence, the tipping pulse is typically90°, but the refocusing pulses are less than 180°. This is in contrastto the CPMG sequence in which the refocusing pulses are 180° pulses.

Generally, the geometry of the NMR measurement device gives rise to avolume in the earth formation where the B₀ field has the correctstrength to fulfill a resonance condition and in which an RF field canbe presented with a substantial strength and orientation to reorientnuclear spins within the volume. This volume is often referred to as thesensitive volume. For a tool in motion, as the tool moves axially, thevolume containing those protons excited by the excitation pulse (firstpulse of the echo sequence) moves away from the sensitive volume. Hence,the number of spins available to contribute to the subsequent NMR signalis reduced with each subsequent echo. As a consequence, those echoesobtained later in an echo sequence with axial tool motion appear smallcompared to those echoes obtained later in an echo sequence acquiredwith no tool motion. “Later echoes” does not mean that only the lastechoes of a sequence are affected. In fact, the loss of signal startsright at the beginning of a sequence and develops over time in a uniquepattern.

The magnet configuration of FIG. 2 produces a somewhat inhomogeneousstatic magnetic B₀ field. Measured in the axial direction, this fieldhas a minimum at the center of the NMR sensor and increases in magnitudeto a maximum at the magnets. The result of this configuration on avolume of formation being traversed in an axial direction is that duringconstant axial motion the formation first comes close to one of themagnets and is magnetized by this higher field. As the NMR sensor centermoves closer, the effective B₀ field decreases. But the formation“remembers” the earlier higher magnetization and only gradually decays,with the time constant T₁, towards the minimum equilibrium magnetizationB₀ field located in the center.

In general, NMR echo sequences are repeated several times for thepurpose of increasing the final signal-to-noise ratio. Even withoutconcern over signal-to-noise ratio, an echo sequence is usually repeatedat least once in order to form a phase-alternated pair (PAP) for thepurpose of removing offset and ringing effects.

At the end of a sequence obtained with axial tool motion, themagnetization of the sensitive volume is substantially zero. A wait timeduring which re-magnetization of the formation occurs is used as part ofthe sequence of pulses. Choosing a wait time of at least 5 times thelongest T₁ of the formation ensures that the formation is fullymagnetized (>99% magnetization) immediately prior to the excitationpulse of the ensuing sequence. However, shorter wait times are oftenchosen in order to achieve a higher NMR data rate, leading to animproved axial resolution or signal-to-noise ratio. The drawback ofshortening TW is that the formation may not be fully magnetizedimmediately prior to the ensuing sequence. As a consequence, the totalporosity that is measured in a tool having axial motion can be too low,and the measured T₂-distribution is generally distorted, mainly for thelonger T₂ components.

The method of the present invention corrects for artifacts that resultfrom axial motion and from a shortened wait time (TW) betweenconsecutive pulse sequences. The correction of the spin echo decay (andhence T₂ distribution) for axial motion is referred to herein asCorrection A and the correction for premagnetization and shortened TW isreferred to herein as Correction B.

The simulations of FIGS. 4-8 are obtained using an NMR simulation for atool such as that shown in FIG. 2. For these simulations, the durationof the applied excitation pulse is 50 μs, and the duration of theapplied refocusing pulses is 70 μs. Such a pulse sequence has beendescribed in U.S. Pat. No. 6,466,013 to Hawkes et al., having the sameassignee as the present application and the contents of which areincorporated herein by reference. It may be referred to in the presentdocument as the Optimized Rephasing Pulse Sequence (ORPS). For thesimulation, a pulse amplitude of 40 A is used. The sensitive volume issubstantially located within radial distances r=130 mm to r=230 mm andwithin axial distances z=−70 mm to z=70 mm. A 2-dimensional simulationis used.

FIG. 4 shows six echo trains obtained at various axial velocities of adrillstring through a formation. The six echo trains are obtained atdifferent velocities: (402), (404), (406), (408), (410) with increasingvelocity, (400) with zero velocity. Time is measured along thehorizontal axis and normalized NMR signal magnitude along the verticalaxis. 1000 echoes are used in the pulse sequence with TE=0.6 msec.Significantly, the curves obtained at the smaller non-zero velocities(402, 404, 406, and 408) can all be derived from the curve obtained forthe highest velocity (410) by simply compressing the horizontal axis.

The results of this compression of the horizontal axis is shown in FIG.5. The echo decay curves obtained with axial motion (402, 404, 406, 408,and 410 in FIG. 4) fall now nearly on top of each other, as indicated bythe curve 502. The curve 500 indicates a signal obtained with no axialtool motion. This alignment of “compressed” echo decays indicates thatthe echo amplitudes are not velocity-dependent, but onlyposition-dependent. In other words, there is no significant phase errordue to the magnitude of the axial motion. This finding is consistentwith the fact (not shown) that there is no significant change in theimaginary part of the NMR signal when comparing zero velocity and thehighest velocity.

A first correction, referred to hereafter as Correction A is discussednext. This correction is intended to address the effects of axial toolmotion. The curves of FIG. 4 can be fitted to a mathematical function.Due to the characteristic shape at the start of the echo sequence, asimple polynomial does not work well. However, a perturbing term can beintroduced to characterize the fluctuation at small times. An exemplaryfunction including a damped cosine term as the perturbing term isemployed for the example of Eq. (3).ƒcA(t)=P ₀ +P ₁ ·t+P ₂ ·t ² +P ₃ ·t ³ +P ₄ ·t ⁴ +P ₅ ·t ⁵ +P ₆ ·t ⁶ +P ₇e ^(−P) ⁸ ^(−t) ·cos(P ₉ ·t)  (3).Eq. (3) adequately fits the echo decay curve with the highest velocity(410 of FIG. 4). P-parameters can be obtained through a fitting methodto the raw data. It should be noted, that the equation strongly dependson the geometric shape of the sensitive area, where the measurement iscarried out. The use of a damped cosine in Eq. (3) is not meant as alimitation of the present invention, and any function appropriate forthis mathematical description can be used.

Assuming an array of NMR echo amplitudes A(t) and of axial velocityν_(axial), one can calculate the corrected amplitude A_(cor)(t) usingthe formula shown in Eq. (4) below: $\begin{matrix}{{A_{cor}(t)} = \frac{A(t)}{f_{c}\left( {t \cdot \frac{v_{axial}}{v_{ref}}} \right)}} & (4)\end{matrix}$where ƒc is the function expressed in Eq. (3) with the parameter setgiven there. This correction can be used for any NMR channel, i.e.,independently, real and imaginary (or in-phase and quadrature).

FIG. 6 shows simulations obtained assuming an infinite T₂ decay. Rawtraces from the simulations obtained at different axial velocities areshown. For convenience, the different axial velocities will be termed aslow velocity, medium velocity, medium high velocity, and high velocity,In FIG. 6, these are shown by 601 b, 602 b, 606 b), and 610 brespectively The traces which result after applying Correction A foraxial velocity are labeled (601 a), (602 a), (606 a), and (610 a). Thehorizontal axis displays the number of echoes, while the vertical axisdisplays normalized amplitude. In order to better differentiate theindividual curves vertical offsets are applied to the curves 601 a/b,602 a/b and 606 a/b. Ideally, the corrected echo traces should be flatlines. As FIG. 6 shows, the corrected traces substantially approach theideal.

Variations in the shapes of RF pulses affect the correspondingartifacts. The exact shape of the motion artifact depends on the type ofRF pulse sequence used. FIG. 7 shows simulations using a different RFcurrent amplitude. Raw traces from the simulations were performed at thedifferent axial velocities from above. Low velocity (702 b), medium low(704 b), medium (706 b), medium high (708 b), and high velocity (710 b)are shown. The traces which result after applying Correction A for axialvelocity are labeled for (702 a), (704 a), (706 a), (708 a), and (710a). The horizontal axis displays the number of echoes, while thevertical axis displays normalized amplitude. In order to betterdifferentiate the individual curves vertical offsets are applied to thecurves. There are 1000 echoes shown with inter-echo spacing TE=0.6 msec.Through comparison of FIG. 7 to results shown in FIG. 6 for the higherRF-pulse amplitude (on which the correction function is based),Correction A proves to be robust under variations of the RF pulse.

Alternately, varying the ratio of RF pulse areas between the excitationpulse and refocusing pulses leads to various artifacts. For instance,for a sequence with a selective excitation pulse, the signal reductionat the beginning of the sequence can be avoided. A correction functioncan be chosen so as to enable correction of a trace resulting from theselective excitation pulse sequence. Alternatively, a different fitfunction can be chosen for different RF pulse sequences or differentecho integrations, such as disclosed in U.S. patent appl. Ser. No.10/839,478 of Blanz et al.

FIG. 8 shows that the correction function of Eq. (1) obtained using asimulation with infinite T₂ can be used for real NMR signals havingfinite T₂. A thousand echoes are shown along the horizontal axis at aninterecho spacing of TE=0.6 ms. The vertical axis shows NMR amplitudesbut with vertical offsets to differentiate the curves on the graph.Corrections are made on curves 840 and 885. Simulations are performedusing a finite T₂ (T₂=1 sec) and different velocities for (840) and(885). Three echo decays are shown for each velocity: the uncorrecteddecay, the corrected decay, and the zero velocity decay. For example, inFIG. 8 uncorrected decay 840 is corrected to obtain corrected decay 805,which is compared with zero velocity decay 800. Similarly, uncorrecteddecay 885 is corrected to obtain corrected decay 880, which is comparedwith zero velocity decay 860.

Another correction, referred to as Correction B, that may be applied inthe present invention to correct for effects of excessivepre-magnetization and reduced TW, is discussed next. FIG. 9 shows 2 ORPSsequences only separated by a wait time TW. Both ORPS have 1000 echoeseach, TE=0.6 s. The wait time TW is 1 s.

When the wait time between ORPS sequences is shorter than five times thelongest T₁ in the earth formation, the latter sequence begins beforeproper magnetization has been achieved. Therefore, the amplitude of theresultant NMR signal depends on the degree of remnant magnetizationafter the previous sequence and on the duration of the wait time. Themagnetization after an ORPS sequence applied with axial tool motion issubstantially zero. However, if there is no motion during the firstORPS, an appreciable amount of z-magnetization is left. FIG. 10 showssimulation examples of echo decays produced using the second ORPSsequence from FIG. 9. For the simulation of FIG. 10, T₁=T₂=2.5 sec.Simulations are made at zero velocity (1000), low axial velocity (1021),increasing axial velocities (1040), (1062), (1081), (1102) and highestvelocity (1120). Excessive premagnetization effects due to movingformation material, which has been exposed to a higher magnetic field inthe close vicinity of the magnets, can be seen. In this case the initialmagnitudes (at t=0 ) of the decay traces of FIG. 10 increase withincreasing velocity—with one exception. An exception is found in thetrace of zero velocity, which has an initial amplitude higher than theinitial amplitudes obtained at higher velocities. This exception is highdue to z-magnetization left after the end of the first ORPS, which isdue to the periodic nature of ORPS. Axial velocity disturbs thiscoherence effect.

In order to counteract the anomalous initial amplitude found at zerovelocity, a shortened saturation sequence can be applied. An exemplaryshortened saturation sequence is an aperiodic pulse sequence (APS) suchas shown in FIG. 11.

In an exemplary mode of the present invention, an APS is constructedwith eight excitation pulses (e.g. of 50 μs length) with reducinginterpulse times 6400 μs, 3200 μs, 1600 μs, 800 μs, 400 μs, 200 μs, 100μs. The corresponding phases are 0°, 180°, 90°, 270°, 0°, 180°, 90°, and270°. The total duration of this sequence is 12.7 ms+8*50 μs=13.1 ms.

For reliable pre-magnetization correction (B) a short aperiodicsaturation sequence as described herein is preferable, not only for thesimulation but also in the real NMR logging run. As an eight-pulse APSis only 13 ms long, there is no disadvantage in doing this. This isshown in FIG. 11 by the initial ORPS (or CPMG sequence) 1121, the APSsequence 1125, the wait time 1125, and a repeat of the ORPS 1127.

FIG. 12 shows results of simulations to obtain corrections for excessiveprepolarization and long T₁ (with insufficient wait time). Thesesimulations employ an APS and have the following properties: the firstORPS has 1000 pulses with TE=0.6 ms; TW=6 sec; and the second ORPS hasonly two refocusing pulses and produces two echoes of which the secondecho is used for determination of the echo amplitude. Therefore FIGS. 12does not show a discontinuity towards zero velocity.

The magnitude of the echo amplitude obtained using the pulses sequenceof FIG. 11 depends on the two variables, axial velocity (v), and T₁.Analysis using discrete values of T₁ can be performed. In an example ofa complete analysis, 6 discrete velocities and 6 discrete values of T₁are chosen and a simulation is run for each combination of these. Thisgives rise to a matrix of normalized echo amplitudes S_(n) such as shownin the matrix of Eq. (5). $S_{n} = \begin{bmatrix}1 & S_{12} & \cdots & S_{1M} \\1 & S_{22} & \cdots & S_{2M} \\\vdots & \vdots & ⋰ & \vdots \\1 & S_{N2} & \cdots & S_{NM}\end{bmatrix}$From left to right in matrix S_(n), T₁ increases from zero to a maximumvalue. From top to bottom, the axial velocity increases from zero to amaximum value. A graphical 3-dimensional representation of matrix S_(n)is shown in FIG. 12.

The simulated echo amplitudes of matrix Sn can be fitted to ananalytical function. In one embodiment of the invention the fittingfunction is a polynomial of the form $\begin{matrix}{{p\left( {v,t} \right)}:={\sum\limits_{i = 0}^{{last}{({coeffs})}}{{coeffs}_{i} \cdot v^{I_{i,0}} \cdot t^{I_{i,1}}}}} & (6)\end{matrix}$

where v is the axial velocity and t the spin-lattice relaxation time T₁with the exponents taken from the Table 1 below. TABLE 1 I I_(i, 0)I_(i, 1) 0 1 2 1 0 3 2 0 2 3 0 1 4 1 1 5 2 1 6 0 0 7 1 0 8 2 0 9 3 0Because the fit function necessarily depends on two variables, amultivariate regression can be used. Eqs. (7) below shows fit matrixS_(fit) and and the form of the error matrix (using a polynomial of thethird order). $\begin{matrix}{{S_{fit} = \begin{bmatrix}S_{f11} & S_{f12} & \cdots & S_{f1M} \\S_{f21} & S_{f22} & \cdots & S_{f2M} \\\vdots & \vdots & ⋰ & \vdots \\S_{fN1} & S_{fN2} & \cdots & S_{fNM}\end{bmatrix}}{{100 \cdot \left( {S_{fi} - S_{n}} \right)} = \begin{bmatrix}S_{e11} & S_{e12} & \cdots & S_{e1M} \\S_{e12} & S_{e22} & \cdots & S_{e2M} \\\vdots & \vdots & ⋰ & \vdots \\S_{eN1} & S_{eN2} & \cdots & {Se}_{NM}\end{bmatrix}}} & (7)\end{matrix}$

FIG. 13 shows the percentage error between the fit and simulated data(S_(fit)−S_(n))*100. An improved fit can be obtained through use of ahigher-order polynomial in exchange for the inconvenience of an increasein the number of coefficients.

Along with axial velocity (v) and T₁, a third parameter, TW, can also bevaried, and a polynomial of three variables, p(ν, T1, TW), can be fittedto the resultant curve. The two-dimensional matrix of equation 5 andFIG. 12 then becomes a 3-dimensional matrix, with the third dimensionbeing the variable TW. The power of exponents (Table 3) will then havethree columns instead of two. For the same order polynomial (here thirdorder), there will be more coefficients. For a third dimension, thereare 20 coefficients, while for two dimensions there are only 10coefficients. Rather than plotting one graph, equivalent to FIGS. 12-13,several graphs can be plotted for individual TWs.

To apply correction B (to correct for artifacts due to premagnetisationand shortened TW), one can divide the echo decay amplitudes (preferablyafter having applied correction A) by the scalar resulting fromevaluation of the polynomial outlined in Eq. (6). The axial velocity (v)and T₁ of the NMR sample are recorded prior to this correction.

Correction B can be used with any echo decay sequence. The success ofthis method of using Correction B is due to the fact that Correction Bonly corrects the magnetization at the beginning of the echo sequence.This initial magnetization depends on the (magnetic) geometry of the NMRlogging tool, the axial velocity (v) during the wait time TW, and T1.

Correction B is an approximation for low axial velocities. At high axialvelocities, the z-magnetization is affected by relaxingpre-magnetization also within the ORPS sequence. Including the effectsof high axial velocity in the pre-magnetization involves increasing thedimension of the polynomial by 1, i.e. the use of a polynomial of 3 or 4variables.

FIG. 14 shows a flowchart of the present invention in the presence of aT₁ distribution. In a real earth formation, a T₁ distribution (ratherthan a single T₁ value) should be expected. Also a T₂ distribution isexpected. After T₂ inversion of the echo envelope data (after havingapplied correction A), each T₂ bin represents a single exponential decayfunction with a characteristic T₂ and the weight given by the height ofthe bin. The superposition of all these exponential functionsconstitutes a multiexponential fit to the original decay data. Becauseof the linear addition of individual exponential decays, Correction Bcan be applied to each exponential separately, i.e. applying CorrectionB to each bin of the T₂ distribution, where all bins have a common axialvelocity but individual T₁. For the case of T₁=f(T₂) where the functionf(T₂) is known, the required T₁ can be obtained directly from thedistribution of T₂. For example, the relationship between T₁ and T₂might be simply a factor T₁=fac*T₂ were “fac” would normally be in therange of 1 to 2. Making use of this relationship enables one toattribute a specific T₁ to each bin of the T₂ distribution. Correction Bcan therefore be applied in the form of a height correction factorindividually to each bin to arrive finally at the true T₂ distribution,where “true” means that all artifacts due to (constant) axial velocityand insufficient wait time are corrected.

In Box 1501 of FIG. 14 the raw data is obtained, typically using a phasealternated pair sequence. Correction A (correction for axial velocity)is applied in Box 1503. Any desired phase correction that is needed canbe applied in Box 1505. A T₂ inversion can be obtained having n bins(Box 1507). As T₁ is functionally dependent on T₂, the T₁ correspondingto each T₂ is consequently calculated in Box 1509. Correction B(correction dependent on axial velocity and T₁) is applied to each T₂bin in Box 1511. The correction B is applied to this decayingexponential. From the application of correction B, a corrected T₂ isobtained for each bin, and thence for the entire T₂ distribution (Box1513). The results of the calculations leads to improved results fortrue total porosity (Box 1515).

FIG. 15 shows a simulation of NMR data using the APS sequence of FIG.11. A simulation is performed for a low drilling speed and T₁=2.2 msecto test the corrections A and B for an exemplary case. Trace 1600 is thereference echo decay trace obtained with full equilibrium magnetizationand zero velocity. Trace 1620 is an uncorrected echo decay trace. Trace1610 is the trace resulting after correction A is applied to trace 1620,and trace 1605 is the trace resulting after correction A and B areapplied. This data is not normalized. The corrected trace 1605 is about1% lower than the reference trace 1600. The percentage error just statedfor FIG. 15 is a relative error. At a total porosity of (e.g.) true 20%,a relative error of −1% would lead to a result that is by 1% (relative)too small, i,e this porosity would be plotted as 20%*0.99=19.8%.

Pulse sequences or echo processing methods that are a priori lesssensitive to motion than standard pulse sequences can be used with thepresent invention. Some such methods have been disclosed in U.S. patentappl. Ser. No. 10/839,478 of Blanz et al. Such pulse sequences are easyto use (at the penalty of some loss of signal-to-noise ratio). Methodsto reduce the sensitivity to irregular small amplitude motion(vibration) can be combined with the corrections A and B for ROPcorrection as described in this report. Correction A will depend on thisother motion artifact reduction method and must be tailored accordingly.

The invention has been described with reference to a NMR device that ispart of a BHA conveyed on a drillstring. The invention is equallyapplicable for NMR devices conveyed on coiled tubing, wireline, andslickline. The processing described herein may be done using a downholeprocessor and the results stored on a suitable memory downhole ortelemetered to the surface. Alternatively, the data may be stored on adownhole memory and processed when the BHA is tripped out of theborehole. With improved telemetry capability, it should be possible totelemeter the NMR measurements to a surface location and do theprocessing there.

While the foregoing disclosure is directed to the specific embodimentsof the invention, various modifications will be apparent to thoseskilled in the art. It is intended that all such variations within thescope and spirit of the appended claims be embraced by the foregoingdisclosure.

1. A method of processing nuclear magnetic resonance (NMR) signals froman earth formation, the method comprising: (a) conveying a NMR loggingtool into a borehole in said earth formation and moving it axiallytherein with an axial velocity; (b) polarizing nuclear spins in saidearth formation; (c) pulsing an antenna on said NMR logging tool with atleast one pulse sequence and producing spin echo signals, said at leastone pulse sequence including an excitation pulse and a plurality ofrefocusing pulses; and (d) correcting said spin echo signals using afunction of said axial velocity and producing first corrected spin echosignals.
 2. The method of claim 1 wherein said logging tool is conveyedinto said borehole on one of (i) a wireline, and, (ii) a slickline,(iii) a drillstring, and, (iv) coiled tubing.
 3. The method of claim 1polarizing said spins further comprises producing a static magneticfield in said earth formation using a magnet on said logging tool. 4.The method of claim 3 wherein a carrier frequency of said at least onepulse sequence is related to a field strength of said static magneticfield.
 5. The method of claim 1 wherein said correcting furthercomprises scaling said spin echo signals by a normalizing functionrelated to said axial velocity and a reference velocity.
 6. The methodof claim 5 wherein said correcting further comprises using an equationof the form${A_{cor}(t)} = \frac{A(t)}{f_{c}\left( {t \cdot \frac{v_{axial}}{v_{ref}}} \right)}$wherein A (t) is one of (i) an in-phase component of said spin echosignals, (ii) a quadrature component of said spin echo signals, and,(iii) an amplitude of said spin echo signals, t is a time, A_(cor)(t) isa corrected signal, ν_(axial) is said axial velocity, and ν_(ref) issaid reference velocity, and ƒ_(c) is said normalizing function.
 7. Themethod of claim 6 wherein said function ƒ_(c) comprises a polynomialfunction and a perturbing term.
 8. The method of claim 7 wherein saidperturbing term comprises a damped oscillating term.
 9. The method ofclaim 1 wherein said excitation pulse has a tip angle that issubstantially equal to 90°.
 10. The method of claim 1 wherein said atleast one pulse sequence comprises a plurality of pulse sequences with await time TW between an ending time of a pulse sequence and a startingtime of a subsequent pulse sequence.
 11. The method of claim 10 furthercomprising using a saturation sequence after an ending time of a pulsesequence of said plurality of pulse sequences.
 12. The method of claim11 wherein said correcting further comprises scaling said spin echosignals by a normalizing function related to said axial velocity, alongitudinal relaxation time of said earth formation, a referencevelocity and a wait time.
 13. The method of claim 10 wherein saidcorrecting further comprises scaling said spin echo signals by anormalizing function related to said axial velocity, a longitudinalrelaxation time of said earth formation, a reference velocity and a waittime.
 14. The method of claim 1 wherein said correcting furthercomprises scaling said spin echo signals by a normalizing functionrelated to said axial velocity, a longitudinal relaxation time of saidearth formation, a reference velocity and a wait time.
 15. The method ofclaim 12 wherein said normalizing function comprises a polynomialfunction of said longitudinal relaxation time and said axial velocity.16. The method of claim 1 wherein said logging tool is conveyed in saidborehole on a bottom hole assembly (BHA), the method further comprisingdetermining said velocity using measurements by an accelerometer on saidBHA.
 17. The method of claim 1 wherein said nuclear spins arecharacterized by a transverse relaxation time (T₂) distribution and alongitudinal relaxation time (T₁) distribution, the method furthercomprising estimating a porosity of said earth formation, said estimateof porosity based in part on said axial velocity, an estimate of said T₂distribution and an estimate of said T₁ distribution
 18. The method ofclaim 17 wherein said estimation of said porosity further comprisesapplying a phase correction to said first corrected spin echo signals togive second corrected spin echo signals.
 19. The method of claim 18further wherein said estimation of said porosity further comprisesinverting said second corrected spin echo signals to give a distributionof said transverse relaxation times T₂, said distribution defined over aplurality of bins.
 20. The method of claim 19 wherein said estimation ofsaid porosity further comprises determining from said T₂ distribution adistribution of T₁.
 21. An apparatus for Nuclear Magnetic Resonance(NMR) logging of an earth formation, the apparatus comprising: (a) a NMRlogging tool conveyed into a borehole in said earth formation by aconveyance device and moved axially therein with an axial velocity; (b)a magnet on said NMR logging tool which polarizes nuclear spins in saidearth formation; (c) an antenna on said logging tool which is pulsedwith at least one pulse sequence and produces spin echo signals fromsaid nuclear spins, said at least one pulse sequence including anexcitation pulse and a plurality of refocusing pulses; (d) a receiverwhich receives said spin echo signals; and (e) a processor whichcorrects said spin echo signals using a function of said axial velocity,and produces first corrected spin echo signals.
 22. The apparatus ofclaim 21 wherein said conveyance device is selected from the groupconsisting of (i) a wireline, and, (ii) a slickline, (iii) adrillstring, and, (iv) coiled tubing.
 23. The apparatus of claim 21wherein a carrier frequency of said pulse sequence is related to a fieldstrength of a static magnetic field produced by said magnet.
 24. Theapparatus of claim 21 wherein said processor corrects said spin echosignals using a normalizing function related to said axial velocity anda reference velocity.
 25. The apparatus of claim 24 wherein saidprocessor corrects said spin echo signals using an equation of the form${A_{cor}(t)} = \frac{A(t)}{f_{c}\left( {t \cdot \frac{v_{axial}}{v_{ref}}} \right)}$wherein A(t) is one of (i) an in-phase component of said spin echosignals, (ii) a quadrature component of said spin echo signals, and,(iii) an amplitude of said spin echo signals, t is a time, A_(cor)(t) isa corrected signal, ν_(axial) is said axial velocity, ν_(ref) is saidreference velocity, and ƒ_(c) is said normalizing function.
 26. Theapparatus of claim 21 wherein said excitation pulse has a tip angle thatis substantially equal to 90°.
 27. The apparatus of claim 21 whereinsaid at least one pulse sequence comprises a plurality of pulsesequences with a wait time TW between an ending time of one of saidplurality of pulse sequences and a starting time of a subsequent pulsesequence.
 28. The apparatus of claim 27 wherein said antenna furtherpulses said earth formation with a saturation sequence after an endingtime of a pulse sequence of said plurality of pulse sequences.
 29. Theapparatus of claim 28 wherein said processor further scales said spinecho signals by a normalizing function related to said axial velocity, alongitudinal relaxation time of said earth formation, and a referencevelocity.
 30. The apparatus of claim 21 wherein said conveyance deviceis a bottom hole assembly (BHA)
 31. The apparatus of claim 30 furthercomprising an accelerometer on said BHA, and wherein said processor usesmeasurements made by said accelerometer for determining said axialvelocity.
 32. The apparatus of claim 21 wherein said nuclear spins arecharacterized by a transverse relaxation time (T₂) distribution and alongitudinal relaxation time (T₁) distribution, and wherein saidprocessor further estimates a porosity of said earth formation, based inpart on said velocity, an estimate of said T₂ distribution and anestimate of said T₁ distribution
 33. The apparatus of claim 32 whereinsaid processor further estimates said porosity by applying a phasecorrection to said first corrected spin echo signals to give secondcorrected spin echo signals.
 34. The apparatus of claim 33 wherein saidprocessor further estimates said porosity by inverting said secondcorrected spin echo signals to give a distribution of said transverserelaxation times T₂, said distribution defined over a plurality of bins.